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The exterior derivative $d:\mathcal{A}^1(M)\to\mathcal{A}^2(M)$ can be regarded as an connection on $T^*M\to M$. If $g$ is a Riemannian connection on $M$, we can can pull $d$ back to get an connection $\nabla$ on $TM\to M$, explicitly $$\nabla_XY:=\left(i_X(d(Y^\flat))\right)^\sharp,$$ where $TM\xrightarrow{\flat}T^*M\xrightarrow{\sharp}TM$ denote the canonical isomorphisms induced by $g$ and $i_X:\mathcal{A}^k(M)\to\mathcal{A}^{k-1}(M)$ is the contraction with $X$. One would expect $\nabla$ to be exactly the Levi-Civita-Connection, however I have calculated its Christoffel symbols as $$\Gamma_{ij}^k=\frac{1}{2}g^{lk}\left(\partial_ig_{jl}-\partial_lg_{ji}\right),$$ so we are missung missing the $\partial_j g_{il}$ summand inside the brackets. Did i make some stupid mistake or do we really have "yet another canonical connection" on $M$?!


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I just found my stupid mistake: $\omega\mapsto d\omega$ is no connection. – Robert Rauch Feb 29 '12 at 17:41
It is a theorem that the L-C connection is the unique canonical connection on a Riemannian manifold. This is done in the book Natural operations in differential geometry by Ivan Kolar, Jan Slovak and Peter W. Michor, which you can get (legally!) online. – Mariano Suárez-Alvarez Feb 29 '12 at 18:27

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